Updated 4 June 2017
Given that this question/answer have gained some popularity, I figured it was worth an update.
When this question was originally posted, MySQL had no support for JSON data types and the support in PostgreSQL was in its infancy. Since 5.7, MySQL now supports a JSON data type (in a binary storage format), and PostgreSQL JSONB has matured significantly. Both products provide performant JSON types that can store arbitrary documents, including support for indexing specific keys of the JSON object.
However, I still stand by my original statement that your default preference, when using a relational database, should still be column-per-value. Relational databases are still built on the assumption of that the data within them will be fairly well normalized. The query planner has better optimization information when looking at columns than when looking at keys in a JSON document. Foreign keys can be created between columns (but not between keys in JSON documents). Importantly: if the majority of your schema is volatile enough to justify using JSON, you might want to at least consider if a relational database is the right choice.
That said, few applications are perfectly relational or document-oriented. Most applications have some mix of both. Here are some examples where I personally have found JSON useful in a relational database:
When storing email addresses and phone numbers for a contact, where storing them as values in a JSON array is much easier to manage than multiple separate tables
Saving arbitrary key/value user preferences (where the value can be boolean, textual, or numeric, and you don’t want to have separate columns for different data types)
Storing configuration data that has no defined schema (if you’re building Zapier, or IFTTT and need to store configuration data for each integration)
I’m sure there are others as well, but these are just a few quick examples.
If you really want to be able to add as many fields as you want with no limitation (other than an arbitrary document size limit), consider a NoSQL solution such as MongoDB.
For relational databases: use one column per value. Putting a JSON blob in a column makes it virtually impossible to query (and painfully slow when you actually find a query that works).
Relational databases take advantage of data types when indexing, and are intended to be implemented with a normalized structure.
As a side note: this isn’t to say you should never store JSON in a relational database. If you’re adding true metadata, or if your JSON is describing information that does not need to be queried and is only used for display, it may be overkill to create a separate column for all of the data points.